Question: A bag contains $4$ red balls, $3$ green balls, and $5$ blue balls. If we choose a ball, then another ball without putting the first one back in the bag, what is the probability that the first ball will be blue and the second will be blue as well? Write your answer as a simplified fraction.
Solution: The probability of event A happening, then event B, is the probability of event A happening times the probability of event B happening given that event A already happened In this case, event A is picking a blue ball and leaving it out. Event B is picking another blue ball. Let's take the events one at at time. What is the probability that the first ball chosen will be blue? There are $5$ blue balls, and $12$ total, so the probability we will pick a blue ball is $\dfrac{5} {12}$ After we take out the first ball, we don't put it back in, so there are only $11$ balls left. Also, we've taken out one of the blue balls, so there are only $4$ left altogether. So, the probability of picking another blue ball after taking out a blue ball is $\dfrac{4} {11}$ Therefore, the probability of picking a blue ball, then another blue ball is $\dfrac{5}{12} \cdot \dfrac{4}{11} = \dfrac{5}{33}$